Associated quantum vector bundles and symplectic structure on a quantum plane

TL;DR

This paper develops a quantum analogue of associated vector bundles and symplectic structures on a quantum plane, enabling the formulation of classical mechanics concepts in a quantum geometric setting.

Contribution

It introduces a new quantum generalization of associated vector bundles and constructs a symplectic structure on a quantum plane, extending classical geometric and mechanical frameworks.

Findings

Defined a quantum algebra of functions over associated bundles.

Constructed a differential complex for the quantum phase space.

Established a quantum symplectic form and Poisson brackets.

Abstract

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant. We apply this construction to the case of the finite quotient of the SL_q(2) function Hopf algebra at a root of unity (q^3=1) as the structure group, and a reduced 2-dimensional quantum plane as both the "base manifold" and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp_q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do "classical" mechanics on a quantum space, the quantum plane.